Abstract

Atmospheric turbulence is a major limiting factor in an optical wireless communication (OWC) link. The turbulence distorts the phase of the propagating optical fields and limits the focusing capabilities of the telescope antennas. Hence, a detector array is required to capture the widespread signal energy in the focal-plane. This paper addresses the bit-error rate (BER) performance of optical wireless communication (OWC) systems employing a detector array in the presence of turbulence. Here, considering the gamma-gamma turbulence model, we propose a blind estimation scheme that provides the closed-form expression of the BER by exploiting the information of the data output of each pixel, which is based on the singular value decomposition of the sample matrix of the received signals after the code-matched filter. Instead of assuming spatially white additive noise, we consider the case where the noise spatial covariance matrix is unknown. The new method can be applied to either the single transmitter or the multi-transmitter cases. Simulation results for different Rytov variances are presented, which conform closely to the results of the proposed model.

1. Introduction

Optical wireless communication (OWC) is an attractive alternative to radio-frequency (RF) links due to its various advantages. The technical merit of laser communications is derived from the fact that it offers a much higher collimated signal than conventional microwave [1

].This super-collimated beam can result in a terminal design with greatly reduce size, weight and transmitter power. Furthermore, laser communication systems are not susceptible to radio frequency interference and are not subject to frequency or bandwidth regulation.

One of the most significant factors that limit the performance of OWC is the effects of turbulent atmosphere [3

]. The magnitudes of the distortions depend on the strength of turbulence and the length of the optical path. Turbulence-induced random fluctuations of the signal intensity at the receiver aperture generate the corresponding increases in the receiver’s average point-spread function (PSF) in the focal plane [10

]. Hence, in the absence of optical turbulence, a detector with the diffraction-limited spot size is sufficient to collect most of the signal energy. However, in the presence of atmospheric turbulence, optical turbulence distorts the phase of the propagating optical fields and limits the extent of focusing capabilities of the antennas. The focal spot size, which limited by the atmospheric Fried parameter, is much greater than the one of ideal diffraction-limited case. In this regime, to capture the widespread signal energy, a larger size detector array is required [7

]. As we increase the detector dimensions to adapt the system to the energy spread of the received beam, the required field-of-view (FOV) of the receiver also increases. Not surprisingly, due to the larger FOV, some terminals can be within the same FOV of one terminal receiving telescope [12

]. Thus, another significant advantage of the detector-array scheme is the ability to receive a number of different signals simultaneously, because the detector array has a FOV large enough to include multiple terminals, in which signals arriving from different terminals are detected by different pixels. The focused beam location of the received signals is defined by a spatial shift, the shift is determined by the spatial locations of the terminals, and then we can estimate signal and noise level in the each pixel associated with each terminal.

Clearly, a large receiver FOV results in a corresponding increase in the number of background photons received [7

] for the detector-array scheme. This paper focuses on the scenario, one of the important constraints for optical networks, that there is a simultaneous communication from a number of terminals to one master terminal. The dimensions of the detector array located on the master terminal must be made large enough to encompass the signal energy and include multiple terminals. In this scheme, each signal from one of a cluster of terminals is modulated by its own unique pseudo-random (PN) sequence, and a common optical system in the master terminal maps these incoming signals onto a high-density, high-speed 2-D detector array. The problem of detecting the scattered optical fields with the detector array has been addressed in [7

]. However, these papers considered only Gaussian channel approximation, and, unlike here, where the knowledge of the amplitudes of signal and noise across the array is needed. In this paper, a closed-form solution for calculating the bit-error rate (BER) of multiple signals over atmospheric turbulence channels is proposed, which is based on the singular value decomposition [19

] using the sample covariance matrix of the received signals and Meijer G-function. This scheme proposed in this paper simultaneously estimates the signals from multiple asynchronous terminals rather than using a number of independent gimbaled telescopes and greatly reduces the need for precise pointing [13

], and the intrinsic interference-tolerance of CDMA systems allows terminals to send unscheduled traffic at any time without regard for whether other terminals are also transmitting at the same time.

The remainder of the paper is organized as follows. Section 2 describes the system model that we proposed. Numeric results are shown in Section 3. Finally, conclusions are given in Section 4.

2. System model

2.1 Characterization of turbulence-induced optical distortion

Atmospheric turbulence, generated by the temperature and pressure changes in the atmosphere, gives rise to the random fluctuations in the atmospheric refractive index called the optical turbulence. Optical turbulence inflicts the deleterious effects on the propagation optical signals [10

]. The stronger turbulence the atmosphere gets, the smaller ther0will be. In the diffraction-limited case, the incoming plane waves is focused into a focal-spot size of 2.44fλ/D, where f and D represent the focal length and the aperture diameter of the receiving telescope, respectively. However, as a beam propagates through atmospheric turbulence, the focal spot size is limited by the Fried parameterr0, and is given by2.44fλ/r0 [10

], assuming that the collection aperture diameterD>r0. The turbulence causes an increase in the spot size and a random distribution of the signal energy into (D/r0)2spatial modes in the detector plane. In this regime, to capture the same amount of the signal energy as that of the diffraction limited case, a larger detector diameter for collecting the signal is needed.

The effects of the atmospheric turbulence alter the refractive index in a statistical manner due to temperature fluctuations. The associated power spectral density can be described by von Karman spectrum for describing the spatial variations of the refractive index [10

whereκ(0≤κ≤∞) is the spatial frequency, κ0=2π/B0,κm=5.92/b0, B0andb0are outer scale and inner scale, respectively. Cn2is the refractive index structure parameter, which is altitude-dependent. Several models for depicting the structure parameter as a function of wind speed, height, and meteorological parameters have been suggested. One of the most widely used models is the Hufnagel-Valley (H-V) model described by [22

where h is the altitude in meters, v is the wind-speed in m/sec, and A is the value ofCn2(0)at the ground. The typical value ofCn2(0)is1.7×10−14m−2/3. The atmospheric diffraction-limited equivalent pupilr0, as defined by Fried 1966, is related to the integrated structure parameter [10

where x is propagation distance. From (3), we can see that the scheme implemented with the structure parameter determines the turbulence strength within the layer.

For optical wave propagation, what we can obtain is a statistical description of the optical variations, because the inhomogeneity of atmospheric turbulence is random. Based on the form of the extended Rytov theory [10

], the received irradiance I can be expressed as the product of two statistically independent random processesIxandIy, that is I=IxIy, whereIxandIyarise from the large-scale and small-scale turbulent eddies, respectively. Based on the assumption that both the large and small scale effects follow a gamma distribution, the so-called gamma-gamma probability density function (pdf) is therefore derived as [23

2.2 Atmospheric BER Estimator

In this paper, we focus on the scenario that there is simultaneous communication from a number of terminals to one master terminal. For convenience, we investigate the scenario that each of terminal points to the master one, while the signals are received by the latter. Let us assume that the signals from each terminal in the space within the terminal cluster can be a spatial δ-function with temporal-spatial intensity distributionS(R→,t)with R→a point in the space. This situation is formulated as

S(R→,t)=∑l=1LηlSl(t)δ(R→−R⇀l)

(7)

where L is the number of terminal cluster participating in the communication with the same master one. Sl(t)denotes the modulating waveform of the lth terminal, ηlis coefficient including transmitter power, gain, and loss. Then, an optical system located on the master maps these signals onto a high-density, high-speed 2-D detector array [12

]. The array output samples the photoelectrons and processes the signals.

For the lth terminal, the modulating waveform Sl(t)forms through multiplying the spreading waveformal(t)by the data waveformdl(t). Ifαlj,j=0,⋯,N−1, are the elements of the periodic pseudorandom sequence Λl=(αl0,⋯,αl(N−1)) with a period N associated with the lth terminal, i.e., for every j,αlj=αl(j+N), whereαlj(j=0,⋯,N−1) is either 0 or 1, then the spreading waveform al(t) can be expressed as

]. Mathematically, the set (1, 0) is transferred to the set (1,-1) byβlj=2αlj−1, whereβlj=1or−1. So that for any sequenceΛlandΓl, the autocorrelationRΓl,Γl(q)and cross-correlation functionsRΛl,Γl(q)are written, respectively, as

RΓl,Γl(q)={1forq=0−1/Nforq≠0,RΛl,Γl(q)={w/Nforq=00forq≠0

(9)

whereΓl=(βl0,⋯βl(N−1)), andw[=(N+1)/2] is the number of elements forαlj=1within a given code sequence.

, each terminal’s optical transmitter is formed on a cluster of pixels, and each of which contains the signals of the transmitters and whose output photocurrents can be combined to detect the transmitted signals [7

]. Here, the free-space channel from the lth transmitter to the mth pixel can be described as a linear time-invariant attenuation channel having impulse responsehl,min the observation. Hereafter, at pixel m, the received signals can be written as

ym(t)=∑n=−∞∞∑l=1Lsl,n∑j=1Nαl,jhl,m(t−jTc−nT−τl)+nm(t)

(10)

wheresl,nis the signal emitted by the lth terminal at time nT, and T is the bit period. Considering a detector array matched to the receiver FOV consisting of M detector elements, each detector element effectively observes a different spatial mode [7

]. For simplicity of the exposition, we assume that the signals from the terminal l are perfectly synchronized to the detector array. Therefore, we can easily summarize all outlet signals asy(t)=∑m=1Mym(t)+n(t). To obtain a compact representation, we sample the signaly(t)at the chip rateTc. The resulting discrete-time signal component due to the nth symbol interval will bey(n)=[y(n,1),y(n,2),⋯,y(n,N)]T(dim.N×1).

In order to estimate the signal-to-noise ratio (SNR), a code-matched filter containing the specified address code vector is applied toy(n). Then the output vector z at the nth bit can be written aszl=y(n)Γl. The cross correlation betweenΓlandΛlis given by (9), and then, after taking P successive frames of the received signals, theN×Pdata matrix is formed as

]. The covariance ofZlis then derived asO=E[ZlHZl], where (⋅)Hdenotes the conjugate transpose. Without loss of generality, we assume that the received frames and the channel taps associated with the different terminals are uncorrelated and mutually uncorrelated. Thus, O can be decomposed asO=Ol+OI+On, whereOl, OIandOnare the signal covariance matrix, the interference covariance matrix and the noise covariance matrix, respectively. Note that we assume a perfectly synchronous system, so that O would be diagonal, the rank of which is equal to 1. By performing the singular value decomposition of the matrix O [19

In (12), the columns ofUsspan the signal subspace uniquely associated with O, and the columns ofUnspan the noise subspace. The diagonal matrix ∑=diag(ζi) contains the corresponding eigenvalues, which in the nonincreasingly order are given byζ1≥ζ2≥⋯≥ζNP. The eigenvalues of O have the following structure

ζi={σsi2+σI2+σn2i=1,⋯,PσI2+σn2i=P+1,⋯,LPσn2i=LP+1,⋯,NP.

(13)

Thus, the signal subspace can be identified. Using (13), the observation noise, the interference power and the desired signal power are estimated, respectively, as follows

]pe(I)=12erfc(ξ˜l2I), whereerfc(⋅)denotes the complimentary error function. The average BER over the gamma-gamma channel can be obtained by averaging pe(I) over the fading coefficientI, i.e., pe=∫0∞f(I)pe(I)dI.

In order to solve the integral, we express erfc(⋅) in terms of the Meijer G-function [24

Note that Meijer G-function is a standard function and has been embedded into most of the well-known mathematical software packages such as Mathematica and Maple. Thus, using the above closed mathematical forms as shown in (16), we can obtain a criterion to appraise the system performance efficiently.

3. Numerical results

In this section, we show some numerical results using the following parameters:L=4,N=7andλ=850nmfor all figures. The array output located in the master terminal is collected in parallel over the observation time and processed for signal detection. A 256×256-pixel uniform array matched to the FOV of the receiver is used where the distance between adjacent pixels is one half the wavelength. At the transmitter side, a modulator modulated the laser with pseudorandom M-sequence code [25

]. For the reception procedure, the incoming signals, which are the superposition of all the transmitted ones, are collected by a telescope, detected by a detector array, and then processed for the decision.

We will first compare the derived BER based on (16) with the system simulations over the gamma-gamma channel. Figure 2

Fig. 2 BER comparison of the theoretical and simulation results

presents the BER of optical networks as a function of SNR, for three different Rytov variancesσR2of 0.5, 1, and 2. It is clear from Fig. 2 that the derived error probability (plotted in dashed style) provides a good approximation and coincides with that of simulation results for all the cases depicted here, indicating that the estimator is robust to the effects of turbulence when the optical beam is subjected to gamma-gamma irradiance distribution model. Figure 3

Fig. 3 Estimate error versus the received SNR over a chip interval

shows the estimate error versus the received SNR over a chip interval withTCforσR2=0.5. We can see that the system can realize communications when received SNR is smaller than or equal to 0.1, because the system employs CDMA instead of uncoded binary date. Hence, there is no need to adapt the threshold. From Fig. 3, we can clearly see that the algorithm presented dramatically decreases the estimate error when received SNR increases.

To study the effect of the frame-number sampled on estimator performance, in Fig. 4

Fig. 4 Estimated SNR versus the number of frames

, we plot the estimated SNR as a function of the number of frames where actual SNR is 8 dB andσR2is 0.5. The estimated SNR curve (shown with a “+”) is seen to converge to its actual one (shown with an “o”) for a large number of samples, validating the analysis model. Clearly, the deviation of the estimator dramatically decreases as the estimated time increases. We can see that the percentage error falls to below 20% after about 10 frames, and to below 10% after less than 18 frames.

4. Conclusion

In this paper, the computational model of estimating the BER with a detector array for OWC systems over atmospheric turbulence channels is proposed, and performance of this model is examined. In this method, the observation space is separated into two orthogonal subspaces, namely, the signal subspace and the noise subspace, by applying the singular value decomposition on the covariance matrix of the received signals after the code-matched filter. Further, based on Meijer G-function, the closed-form solution of the BER is derived. The comparison between the theoretical and simulation results based on the gamma-gamma channel model indicates that such a model is adequate in estimating the BER.

Acknowledgement

This work is supported by the National Nature Science Foundation of China (60572002).

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